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A class of discrete spectra of non-Pisot numbers. (English) Zbl 1274.11160
The author investigates the class of \(\pm1\) polynomials evaluated at \(q\), and defined as: \[ A(q)=\bigl\{\varepsilon_0+\varepsilon_1q+\cdots+\varepsilon_m q^m:\varepsilon_i\in\{-1,1\}\bigr\}. \] This is usually called the spectrum, and in the thesis of K. G. Hare of 2002 [Pisot numbers and the spectra of real numbers, Ph.d. thesis, Simon Fraser University (2002)] it was proved that, if \(q>1\) is a root of the polynomial \(x^n-x^{n-1}-\dots-x+1\), then \(A(q)\) is discrete. The author here shows a stronger result, namely that, if \(q\) is the greatest real root of the polynomial \(x^n-x^{n-1}-\dots-x^{k+1}+x^k+x^{k-1}+\cdots+x+1\) and \(n>2k+3\), \(k\geq 0\), then \(A(q)\) is discrete, which means that it does not have any accumulation points. The proof is based on a recursive algorithm of D.-J. Feng and Z.-Y. Wen [J. Number Theory 20, 3015–316 (2002; Zbl 1026.11078)] and four lemmas. The author remarks that, if \(n=2k+3\), then the polynomial above can be factored as \((x^{k+1}-x^k-x^{k-1}-\dots-1)(x^{k+2}-1)\). Clearly, in this case the theorem is also valid. This seems to be the limiting case after which the theorem ceases to hold.
11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
11Y60 Evaluation of number-theoretic constants
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